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visits member for 3 years, 9 months
seen Nov 10 at 6:16

Jul
23
comment What are some conceptualizations that work in mathematics but are not strictly true?
@Semiclassical The notion of $\theta^2$ gives me a bit of a headache.
Apr
21
awarded  Commentator
Apr
21
comment Why is P or not P is unsatisifiable by construction?
As for "by construction": the phrase has two related meanings. A "proof by construction" is a proof that asserts an object exists by constructing it. The sentence "By construction, X has property Y" means "the way in which X was described (i.e., its construction) makes it easy to see that it has property Y".
Apr
21
comment Why is P or not P is unsatisifiable by construction?
You're reading an 'or' in the formula where you should be reading an 'and'. In order for a set such as $\Delta$ to be satisfied by a model, every formula in the set has to be satisfied simultaneously by that model. For any model $G$, if $G$ satisfies $\phi[c/u][c'/v]$, then there is some $n$ such that $G$ does not satisfy $\neg \phi_n$. Therefore, it cannot satisfy $\Delta$, since $\neg \phi_n \in \Delta$.
Apr
20
answered Why is P or not P is unsatisifiable by construction?
Apr
3
comment Con(PA) implies consistency of $\mathsf{PA}$ + ¬Con($\mathsf{PA}$)
The one thing I'm a little unclear is what you mean by 'completeness'; if PA is consistent, doesn't that necessarily mean that it's incomplete?
Apr
3
accepted Con(PA) implies consistency of $\mathsf{PA}$ + ¬Con($\mathsf{PA}$)
Apr
2
asked Con(PA) implies consistency of $\mathsf{PA}$ + ¬Con($\mathsf{PA}$)
Aug
21
awarded  Good Question
Feb
16
awarded  Notable Question
Apr
30
accepted Existence of a particular well-ordering of [0,1]
Apr
30
comment Existence of a particular well-ordering of [0,1]
Ah, ok. Thanks!
Apr
30
comment Existence of a particular well-ordering of [0,1]
So I can see how you need CH to show that $\omega_1$ can be bijected with $\mathbb{R}$ (otherwise you just know that $\omega_1$ is uncountable), but I'm not sure how AC comes into play.
Apr
30
comment Existence of a particular well-ordering of [0,1]
I'm familiar with the 'basic' infinite ordinals like $\omega$, less so with $\omega_1$.
Apr
30
comment Existence of a particular well-ordering of [0,1]
Could you explain further?
Apr
30
asked Existence of a particular well-ordering of [0,1]
Apr
26
awarded  Critic
Apr
26
comment Eccentricity of an ellipse
What is C? What is A?
Apr
26
awarded  Supporter
Apr
25
awarded  Yearling